Universal behavior for averages of characteristic polynomials at the origin of the spectrum

Abstract

It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble $\frac{1}{\hat Z_n}|\det M|^{2α}e^{-nV(M)}dM$ of $n\times n$ Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as $n\to\infty$, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.